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title: Computational Geometry Study Notes
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# Texts

- *Discrete and Computational Geometry*, Devadoss

# Reading Notes

## *Devadoss*, 24 Jan 2014

Computational Geometry is *discrete* rather than *continuous*

Fundamental building blocks are the *point* and line *segment*.

*polygon*
: the closed region of the plane bounded by a finite collection of
line segments forming a closed curve that does not intersect
itself. The segments are called *edges* and the points where they meet
are *vertices*. The set of vertices and edges is the *boundary*.

**Theorem 1.1: Polygonal Jordan Curve**
:The boundary $\partial P$ of a polygon $P$ partitions the plane into
two parts.  In particular, the two components of $\Bbb{R}^2\setminus
\partial P$ are the bounded interior and the unbounded exterior.

A point $x$ is *interior* if a ray through it in a fixed direction not
parallel to an edge passes through an odd number of edges.

A point $x$ is *exterior* if a ray through it ... etc ... passes
through an even number of edges.

These two facts form the basis of an algorithm for determining whether
a point is inside a polygon.

*diagonal*
: A line segment of $P$ connecting two vertices and lying in the interior of
$P$, not touching $\partial P$ except at endpoints. Two diagonals are
*noncrossing* if they share no interior points.

*triangulation*
: A decomposition of $P$ into triangles by a maximal set of
noncrossing diagonals. Maximal means that no more diagonals may be
added without crossing.

**Lemma 1.3**
: Every polygon with more than three vertices has a diagonal.

**Theorem 1.4**
: Every polygon has a triangulation.

*polyhedron*
: A 3-d generalization of a polygon, a solid bounded by finitely many
polygons.

*tetrahedron*
: A pyramid with a triangular base. The simplest polyhedron.

Polygon triangularization generalizes to polyhedron
*tetrahedralization*, which is partitioning of the interior into
tetrahedrons whose edges are diagonals of the polyhedron. Not all
polyhedrons can be tetrahedralized!

**Theorem 1.8**
: Every triangularization of a polygon $P$ with $n$ vertices has $n -
2$ triangles and $n - 3$ diagonals.

*ear*
: Three consecutive vertices $a, b, c$ form an *ear* of a polygon if
$a c$ is a diagonal of the polygon. The vertex $b$ is called the ear
*tip*.

**Corollary 1.9**
: Every polygon with more than three vertices has at least two ears.

A vertex of a polygon is *reflex* if its angle is greater than $\pi$
and *convex* if its angle is less than or equal to $\pi$. It is *flat*
if its angle is exactly $\pi$ and *strictly convex* if its angle is
strictly less than $\pi$.

A polygon $P$ is a *convex polygon* if all of its vertices are
strictly convex.

**Lemma 1.18**
: A diagonal exists between any two nonadjacent vertices of a polygon
$P$ if and only if $P$ is a convex polygon.

**Theorem 1.19**
: The number of triangulations of a convex polygon with $n + 2$
vertices is the Catalan number $$ C_n = \frac{1}{n + 1}{n + 1 \choose
n} $$